Properties

Label 720.6.a.be
Level $720$
Weight $6$
Character orbit 720.a
Self dual yes
Analytic conductor $115.476$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [720,6,Mod(1,720)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(720, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("720.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 720.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(115.476350265\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{145}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 45)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{145}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 25 q^{5} + ( - 5 \beta - 40) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 25 q^{5} + ( - 5 \beta - 40) q^{7} + (5 \beta - 400) q^{11} + ( - 20 \beta - 60) q^{13} + ( - 8 \beta + 970) q^{17} + (80 \beta + 756) q^{19} + (126 \beta + 660) q^{23} + 625 q^{25} + (320 \beta + 650) q^{29} + ( - 110 \beta + 2912) q^{31} + ( - 125 \beta - 1000) q^{35} + ( - 360 \beta - 6280) q^{37} + ( - 340 \beta + 200) q^{41} + (320 \beta - 12840) q^{43} + ( - 46 \beta - 9460) q^{47} + (400 \beta - 707) q^{49} + ( - 152 \beta + 24730) q^{53} + (125 \beta - 10000) q^{55} + (245 \beta - 31600) q^{59} + (1360 \beta - 24558) q^{61} + ( - 500 \beta - 1500) q^{65} + ( - 2030 \beta - 3040) q^{67} + (370 \beta - 32600) q^{71} + ( - 1160 \beta + 48870) q^{73} + (1800 \beta + 1500) q^{77} + (670 \beta + 23144) q^{79} + ( - 2682 \beta + 28680) q^{83} + ( - 200 \beta + 24250) q^{85} + ( - 2100 \beta - 43500) q^{89} + (1100 \beta + 60400) q^{91} + (2000 \beta + 18900) q^{95} + ( - 720 \beta + 5090) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 50 q^{5} - 80 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 50 q^{5} - 80 q^{7} - 800 q^{11} - 120 q^{13} + 1940 q^{17} + 1512 q^{19} + 1320 q^{23} + 1250 q^{25} + 1300 q^{29} + 5824 q^{31} - 2000 q^{35} - 12560 q^{37} + 400 q^{41} - 25680 q^{43} - 18920 q^{47} - 1414 q^{49} + 49460 q^{53} - 20000 q^{55} - 63200 q^{59} - 49116 q^{61} - 3000 q^{65} - 6080 q^{67} - 65200 q^{71} + 97740 q^{73} + 3000 q^{77} + 46288 q^{79} + 57360 q^{83} + 48500 q^{85} - 87000 q^{89} + 120800 q^{91} + 37800 q^{95} + 10180 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
6.52080
−5.52080
0 0 0 25.0000 0 −160.416 0 0 0
1.2 0 0 0 25.0000 0 80.4159 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( +1 \)
\(5\) \( -1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 720.6.a.be 2
3.b odd 2 1 720.6.a.y 2
4.b odd 2 1 45.6.a.f yes 2
12.b even 2 1 45.6.a.d 2
20.d odd 2 1 225.6.a.k 2
20.e even 4 2 225.6.b.k 4
60.h even 2 1 225.6.a.r 2
60.l odd 4 2 225.6.b.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.6.a.d 2 12.b even 2 1
45.6.a.f yes 2 4.b odd 2 1
225.6.a.k 2 20.d odd 2 1
225.6.a.r 2 60.h even 2 1
225.6.b.j 4 60.l odd 4 2
225.6.b.k 4 20.e even 4 2
720.6.a.y 2 3.b odd 2 1
720.6.a.be 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(720))\):

\( T_{7}^{2} + 80T_{7} - 12900 \) Copy content Toggle raw display
\( T_{11}^{2} + 800T_{11} + 145500 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T - 25)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 80T - 12900 \) Copy content Toggle raw display
$11$ \( T^{2} + 800T + 145500 \) Copy content Toggle raw display
$13$ \( T^{2} + 120T - 228400 \) Copy content Toggle raw display
$17$ \( T^{2} - 1940 T + 903780 \) Copy content Toggle raw display
$19$ \( T^{2} - 1512 T - 3140464 \) Copy content Toggle raw display
$23$ \( T^{2} - 1320 T - 8772480 \) Copy content Toggle raw display
$29$ \( T^{2} - 1300 T - 58969500 \) Copy content Toggle raw display
$31$ \( T^{2} - 5824 T + 1461744 \) Copy content Toggle raw display
$37$ \( T^{2} + 12560 T - 35729600 \) Copy content Toggle raw display
$41$ \( T^{2} - 400 T - 67008000 \) Copy content Toggle raw display
$43$ \( T^{2} + 25680 T + 105473600 \) Copy content Toggle raw display
$47$ \( T^{2} + 18920 T + 88264320 \) Copy content Toggle raw display
$53$ \( T^{2} - 49460 T + 598172580 \) Copy content Toggle raw display
$59$ \( T^{2} + 63200 T + 963745500 \) Copy content Toggle raw display
$61$ \( T^{2} + 49116 T - 469672636 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots - 2380880400 \) Copy content Toggle raw display
$71$ \( T^{2} + 65200 T + 983358000 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots + 1607828900 \) Copy content Toggle raw display
$79$ \( T^{2} - 46288 T + 275282736 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots - 3349469520 \) Copy content Toggle raw display
$89$ \( T^{2} + 87000 T - 665550000 \) Copy content Toggle raw display
$97$ \( T^{2} - 10180 T - 274763900 \) Copy content Toggle raw display
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